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This is an alternative formulation of the Orthogonal Procrustes problem (consider all the vectors multiplied by the square-roots of the corresponding weights as columns of two matrices with N columns to obtain the alternative formulation). An elegant derivation of the solution on one and a half page can be found in. [3]
The orthogonal Procrustes problem [1] is a matrix approximation problem in linear algebra. In its classical form, one is given two matrices A {\displaystyle A} and B {\displaystyle B} and asked to find an orthogonal matrix Ω {\displaystyle \Omega } which most closely maps A {\displaystyle A} to B {\displaystyle B} .
Matrix-free conjugate gradient method has been applied in the non-linear elasto-plastic finite element solver. [7] Solving these equations requires the calculation of the Jacobian which is costly in terms of CPU time and storage. To avoid this expense, matrix-free methods are employed.
This is applied, e.g., in the Kalman filter and recursive least squares methods, to replace the parametric solution, requiring inversion of a state vector sized matrix, with a condition equations based solution. In case of the Kalman filter this matrix has the dimensions of the vector of observations, i.e., as small as 1 in case only one new ...
The NLEVP collection of nonlinear eigenvalue problems is a MATLAB package containing many nonlinear eigenvalue problems with various properties. [ 6 ] The FEAST eigenvalue solver is a software package for standard eigenvalue problems as well as nonlinear eigenvalue problems, designed from density-matrix representation in quantum mechanics ...
Vectorization is used in matrix calculus and its applications in establishing e.g., moments of random vectors and matrices, asymptotics, as well as Jacobian and Hessian matrices. [5] It is also used in local sensitivity and statistical diagnostics.
Matrix chain multiplication (or the matrix chain ordering problem [1]) is an optimization problem concerning the most efficient way to multiply a given sequence of matrices. The problem is not actually to perform the multiplications, but merely to decide the sequence of the matrix multiplications involved. The problem may be solved using ...
In mathematics, particularly in linear algebra and applications, matrix analysis is the study of matrices and their algebraic properties. [1] Some particular topics out of many include; operations defined on matrices (such as matrix addition, matrix multiplication and operations derived from these), functions of matrices (such as matrix exponentiation and matrix logarithm, and even sines and ...